Papers

Proving Shannon-type entropy inequalities is a fundamental task in information theory that often requires constructing non-trivial linear combinations of known constraints, which is a combinatorial search problem that scales poorly with the number of random variables. We investigate whether small-scale large language models (0.6B–1.7B parameters), fine-tuned on atomic proof steps and combined with guided beam search, can automate this process. On a held-out test set of 60 inequalities spanning n=10 to 15 variables, our 0.6B fine-tuned model achieves an 85% proof success rate with tree search. GPT-5.5 solves 1.7% samples under zero-shot prompting while Psitip solves 33.3% samples. A systematic ablation study across training context length (4096 vs.\ 8192 tokens) and data distribution (n=9-skewed vs not skewed) reveals that a 4096-token not skewed training distribution yields the best performance, with extended context and skewed data providing no marginal benefit. We further identify two dominant failure modes – format failures and step quality degradation – and verify that the beam-scoring heuristic is essential via a controlled ablation (random scoring reduces success from 83% to 23%).

We prove that a finite, connected, singly connected, locally convex higher-rank tree whose $1$-skeleton is planar and which is non-degenerate, in the sense that every edge of each colour forms a commuting square with every other colour, has rank at most four. Under these hypotheses this establishes the planarity conjecture stated in \cite{Pask}. The obstruction side of the argument uses only the non-planarity of $K_5$; it makes no appeal to the four-colour theorem. The engine is a monotonicity property of the set of colours emitted at a vertex (“backward propagation”), which forces, in any finite singly connected non-degenerate $k$-graph, a single vertex emitting all $k$ colours; once $k\ge 5$, local convexity manufactures a subdivision of $K_5$ at such a vertex.

The classical problem of Erdős asks for the minimum number of empty convex $k$-gons determined by an $n$-element point set in the plane. The celebrated empty hexagon theorem, proved independently by Gerken and Nicolás, shows that every sufficiently large planar point set contains a $6$-hole, while Horton's famous construction shows the existence of arbitrarily large point sets with no $7$-hole. In this paper, we initiate the study of the maximum number of $k$-holes in planar point sets with no $(k+1)$-hole. More precisely, for each fixed $k\geq 6$, let $h_k(n)$ be the maximum number of $k$-holes determined by a planar point set in general position, of size at most $n$, and with no $(k+1)$-hole. We prove that there are absolute constants $c_1,c_2>0$ such that $(c_1/k)^{\lfloor k/3\rfloor} n^{\lfloor k/3\rfloor} \leq h_k(n)\leq (c_2/k)^{\lceil k/2\rceil} n^{\lceil k/2\rceil}$.

The mass-critical NLS on Euclidean space $\R^d$ exhibits a strong mass rigidity: all positive ground states are generated from a single profile and have the same ground state mass $\widehat{M}(Q)$. By appealing to bifurcation methods, Dancer constructed in his seminar paper \cite{DancerSolution} solutions to the corresponding equation on $\R^d\times\T$ which decay in the noncompact directions and are nontrivially periodic in one direction. Such bifurcation approach, however, does not provide any energetic characterization of the solutions, and in particular does not explain their relation to the Euclidean ground-states. By introducing a new strict monotonicity mechanism for the prescribed-mass energy level, combining the semivirial-vanishing geometry framework developed in author's recent work, we prove that for any mass $c\in(0,2\pi\widehat{M}(Q))$ the semivirial-vanishing variational problem $m_c$ admits a normalized Dancer-type optimizer which also solves the focusing mass-critical NLS on $\R^d\times\T$. This also gives a sharp complement for the existence results deduced in our earlier work \cite{Luo_LegendreFenchel} via the Legendre-Fenchel duality.

We construct $N=1$ supersymmetric (SUSY) vertex algebras from supersymmetric enhancements of Costello–Gwilliam factorization algebras on super Riemann surfaces. Introducing SUSY factorization algebras defined on embedded SUSY disks together with natural symmetry conditions, we prove a SUSY analogue of the Costello–Gwilliam extraction theorem. As an application, we study the holomorphic sigma model in the BV formalism. For a linear target, we obtain the free $bc$-$\beta\gamma$ system and recover its structure as a SUSY vertex algebra. For general complex targets, we describe the descent of the theory under coordinate changes and identify the resulting SUSY vertex algebra with the chiral de Rham complex. We further show that Ricci-flat Kähler and hyperkähler targets give rise to $N=2$ and $N=4$ supersymmetric enhancements introduced by Ben-Zvi–Heluani–Szczesny.

Bowler et al. introduced the concept of torsoids that describes the 1-separations of a directed graph in a canonical way building on previous progress by Lovász. To fully understand the structure of directed graphs with respect to their 1-separations it remains to find a global structure along which the torsoids are arranged. In this paper, we start the investigation of this global structure for a specific class of directed graphs.

We study statistical inference for deterministic drifts in Gaussian process models under high-frequency observations over an expanding time horizon. Using a least squares-type contrast based on first-order increments, we establish consistency and asymptotic normality under conditions on drift accumulation and increment dependence.A key feature is that the convergence rate is determined jointly by the deterministic signal and the full covariance structure of the weighted Gaussian increments, rather than by local noise roughness alone.For power and fixed-frequency periodic drifts under Gaussian and Ornstein-Uhlenbeck covariance kernels, we derive explicit convergence rates and limiting variances, revealing distinct regimes depending on the drift structure and, for periodic drifts, the noise spectrum. These results clarify the respective roles of sampling frequency and observation horizon.

The weighted difference of squared norms (WDSN) penalty $\ell_1^2-\eta\ell_2^2$ with $0\leq \eta\leq 1$ has attracted considerable attention due to its strong sparsity-promoting ability and favorable reconstruction performance in compressed sensing and inverse problems. However, exact recovery guarantees and restricted isometry property (RIP) analysis for WDSN minimization have not yet been established. In this paper, we address this gap. First, we establish sufficient conditions for the exact recovery of $k$-sparse signals based on the null space property (NSP). Then, under the $\delta_{2k}$-RIP condition, we derive stable recovery guarantees for both $k$-sparse signals and general signals, and characterize upper bounds on the reconstruction error. Furthermore, we propose a WDSN-based regularized model to handle both noiseless and noisy observations in a unified framework. To design an efficient algorithm, we derive an explicit formula for the proximal operator of the WDSN functional. Based on this proximal solver, we develop a suitable variable-splitting scheme within the alternating direction method of multipliers (ADMM) and establish its global convergence under some mild conditions. Finally, numerical experiments show that the proposed method outperforms the iterative half variation method in both noiseless and noisy sparse recovery tasks.

In this paper, we consider MCMC algorithms for Student-$t$ regression models. In three cases, we investigate the efficiency of Markov chains based on the algorithms in terms of whether trace-class results hold or not. First, we consider the case where the parameters follow a matrix-normal-inverse-Wishart distribution and show that the Markov operator associated with a standard data augmentation algorithm is trace-class. Second, we consider the case of an improper prior and univariate outcomes. In this case, the standard Markov operator is not trace-class but the Markov operator associated with a collapsed Gibbs algorithm is trace-class. Third, we consider the case of an improper prior and multivariate outcomes. We obtain a trace-class result for a parameter expanded data augmentation algorithm which is based on a univariate working parameter. Finally, we consider the problem of numerially estimating a convergence rate of the trace-class Markov operator in the second case.

Sparse recovery aims to reconstruct sparse signals from underdetermined and possibly noisy linear measurements. Existing $\ell_{1-2}$ iterative thresholding schemes are first-order methods. We propose an iterative thresholding pursuit method with continuation (ITP-C) for $\ell_{1-2}$-regularized sparse recovery. The method goes beyond first-order thresholding by combining the active-set identification capability of the $\ell_{1-2}$ proximal step with a restricted least-squares pursuit step that provides a second-order update on the identified support. The support is generated adaptively by the thresholding update, and no prior knowledge of the true sparsity level is required. To control the possible instability of the pursuit step while preserving the descent structure of the continuation scheme, we impose a strict descent check with respect to the dynamic objective. We establish convergence of the generated sequence under the Kurdyka-Lojasiewicz framework and prove a local oracle-type property after correct support identification. Numerical experiments on synthetic sparse recovery and image reconstruction illustrate the descent preservation of the proposed safeguard and demonstrate the improved recovery performance of ITP-C over the state-of-the-art baselines.

We investigate the quantum aspects of gauging continuous 1-form global symmetries. In this paper, we study the BV-BRST quantization of a $U(1)$ 2-form gauge field, described geometrically by a $U(1)$ gerbe. Starting from the local Čech data of the gerbe, we construct the corresponding infinitesimal symmetry structure in terms of a Lie 2-algebroid, and show that, together with the associated exact Courant algebroid, it provides a natural geometric framework for the BV-BRST complex of this higher-form gauge theory. In this formulation, the field-ghost tower is encoded directly in the local gerbe data, and the higher Russian formula arises naturally from the relations among the connective structure, the curving, and the 3-form curvature. We further show that the resulting Čech-de Rham bicomplex provides a natural setting for anomaly descent for $U(1)$ 1-form symmetries, and illustrate the construction with explicit examples in Maxwell theory.

This paper considers large-scale T-Sylvester equations of the form $AX - X^\top E^\top + B_1B_2^\top = 0$, which admit a low-rank solution. It is shown that when the unique solution of the T-Sylvester equation is low-rank, the problem naturally reduces to a tangential interpolation problem via oblique projection. The specific interpolation points and tangential directions needed to obtain the low-rank solution are not known a priori, thus requiring an iterative approach. An iterative interpolatory projection algorithm is proposed based on these interpolation conditions, which iteratively refines the interpolation data as the projection matrices expand in the number of columns. Numerical examples demonstrate that the proposed algorithm converges with projection matrices having significantly fewer columns compared to existing Krylov-subspace-based projection methods, confirming the superiority of the proposed algorithm over existing approaches.

All existing convergence rate estimates of Halpern iterations are established only for the case where the anchor coincides with the initial guess. This paper aims to investigate the convergence rate of general Halpern iterations, where the anchor and the initial guess may not necessarily be the same. We present tight convergence rate estimates for both predetermined and adaptive anchoring parameters. These results generalize existing related work.

We present a short proof that the book Ramsey number $R(B_8,B_{10})$ equals 37. The lower bound $R(B_8,B_{10}) \ge 37$ is already available in the literature, so it is enough to rule out a 37-vertex graph containing neither a copy of $B_8$ nor a copy of $B_{10}$ in its complement. The problem as well as the proof were found with AutoMath, an AI-assisted mathematical discovery workflow developed by the first author. A Lean formalization of the upper-bound argument is available in the accompanying repository.

Focused ultrasound (FUS) offers a promising, non-invasive method for modulating neural activity and delivering therapies deep within the brain with immense clinical potential. However, progress in developing transcranial ultrasound (TUS) for clinical applications has been hindered by several factors. The complexity of the human skull causes focal aberrations and attenuation, thereby presenting a major obstacle to the precise targeting of ultrasound waves. Although phased arrays can correct for these aberrations, their high cost and continuous reliance on magnetic resonance imaging (MRI) pose significant obstacles for widespread academic research and clinical translation. To address these challenges, this thesis proposes an innovative framework for the design, registration, and clinical application of acoustic holograms. First, we introduce a novel frequency-domain topology optimization method that overcomes the breakdown of traditional phase-only designs in the megahertz regime by accounting for volumetric wave-propagation effects, thereby achieving high-fidelity focusing. Second, we present a non-invasive registration strategy that utilizes the nonlinear parametric array (PA) effect to enable precise lens alignment without requiring any imaging modalities, such as MRI. Finally, we demonstrate the utility of this nonlinear parametric array (PA) effect as a tool for monitoring ventricular dilation as a non-invasive proxy for intracranial pressure changes in hydrocephalus. Collectively, these developments provide a path toward accessible, high-precision transcranial ultrasound systems for research and clinical use. In addition, we demonstrate a novel platform for in vitro focused ultrasound neuromodulation that leverages acoustics to advance therapeutic discovery.

Suppose that for a mixing $Z^n$-action, $n>1$, there exists a Kirillov-Kushnirenko entropy that is zero for this action and completely positive for its inverse. We prove that this property is generic for the mixing $Z^n$-actions.

Extreme events – such as earthquakes and coronal mass ejections – are common in many chaotic dynamical systems, yet are difficult to characterize and predict due to the subtle instability mechanisms that drive them. In this work, we develop an interpretable technique that reveals the underlying mechanisms behind extreme events and uses them to build data-driven forecasts and intuitive event suppression controllers. In particular, we utilize the covariance balancing reduction using adjoint snapshots (CoBRAS) method to identify linear oblique projections that best capture the sensitivity of a quantity of interest and reconstruct the original state. Importantly, we bypass the need for cumbersome adjoint calculations, instead using backpropagation via modern automatically differentiable numerical frameworks. To accommodate spatially localized events, we also introduce a new variant of CoBRAS to obtain local sensitivity-balanced projections. We demonstrate the utility of this approach to characterize extreme events across a diverse set of challenging systems, including turbulent bursts of energy dissipation in the 2D Kolmogorov Flow, spontaneous synchronization in networks of coupled FitzHugh-Nagumo oscillators, and the localized formation of ocean rogue waves from a modified nonlinear Schrödinger equation. For each example, we show that our simple forecast models accurately predict extreme events and that the underlying mechanisms may be used to design control laws to prevent these events. Finally, we demonstrate that by learning a neural network surrogate model of the dynamics directly from data, we may extend this approach to experimental systems and systems that are not natively written in an automatically differentiable programming language.

We develop parallel and batch-cutting variants of the norm-minimization-based outer approximation algorithm for convex vector optimization. The standard algorithm solves $N_k$ independent subproblems at each iteration~$k$ to evaluate all vertices of the current polyhedral approximation, but processes only the single best cut. We propose two improvements. First, we parallelize the \revise{subproblem evaluations} across $\nw$ workers, reducing per-iteration wall-clock time. Second, we introduce a batch-cutting strategy that adds up to $K$ supporting halfspaces per iteration, using information from all solved subproblems rather than discarding it. We prove that the batch-cutting variant inherits the convergence rate $O(k^{2/(1-q)})$ of the standard algorithm, where $k$ is the number of outer iterations and $q$ is the number of objectives. Computational experiments on eight test problems with $q \in \{2,3,4,5\}$ show that parallelism on 8 cores \revise{increases the speed by a factor of 1.1 to 4.2}, and batch cutting consistently reduces the iteration count by 62–80%. However, the wall-clock benefit of batch cutting is problem-dependent: the additional cuts per iteration accelerate vertex count growth, so batch cutting is most effective when per-vertex subproblem cost dominates.

We establish weighted hp-uniform vertex-patch decompositions in arbitrary space dimension d >= 1 for tensor-product discretizations of H^k-type conforming and nonconforming spaces, with arbitrary fixed Sobolev order k >= 1, on fitted interface meshes. The cells are coordinate-compatible cuboids, the local spaces are Q_{p_K}(K) with arbitrary elementwise degrees satisfying p_K >= 2k-1, and the coefficient may have arbitrarily large jumps across material interfaces. Under local coefficient oscillation bounds and a local high-side connectivity condition, both the conforming H^k space and the nonconforming spaces V_h^{(s)}, 0 <= s <= k, admit stable decompositions with constants which may depend on the fixed parameters d and k, but are independent of the mesh size, all polynomial degrees, neighboring degree ratios, and the global coefficient contrast. The argument combines a Hermite endpoint transform for endpoint jets of order 0,...,k-1, its tensor-product extension, weighted broken patch Poincare inequalities, and a successive correction of normal derivative jumps. Numerical experiments for a three-dimensional DG problem with large coefficient jumps and strongly varying local polynomial degrees support the predicted robustness. For k = 1 the same conclusions hold on uniformly regular mapped cubical meshes whose neighboring element maps agree on each common face.

LLM post-training often relies on reinforcement learning methods that sample multiple rollouts per prompt, yet most existing approaches use a fixed rollout budget for every prompt, despite large differences in the training signal different prompts provide. In this paper, we study adaptive rollout allocation under a fixed global budget and formulate the problem as online resource allocation with prompt-level diminishing returns. Our method, CERO, maintains a Beta posterior over each prompt's success probability and uses the posterior expected Bernoulli variance as a Bayesian estimate of the value of additional rollouts. We use this estimate to construct a concave, saturating utility over cumulative allocations, yielding an objective in which decisions across prompts and epochs are coupled by the global budget. Since the resulting objective is temporally nonseparable, we derive a Fenchel-dual reformulation and update both prompt-level and budget-level dual variables via projected online gradient descent. Under fixed prompt utilities, we prove an $O(\sqrt{K})$ regret bound against the offline allocation benchmark. Experiments on mathematical-reasoning problems show that CERO consistently outperforms GRPO across multiple open-weight LLMs and benchmarks, demonstrating that adaptive rollout budgeting can improve sample efficiency.

This paper investigates a rotatable antenna (RA)-enabled uplink non-orthogonal multiple access (NOMA) system, where a base station equipped with multiple independently RAs serves both ground and aerial users. Specifically, we formulate an energy efficiency (EE) maximization problem by jointly optimizing receive beamforming, user power allocation, and RA rotation. To make the problem tractable, a new block coordinate descent-based algorithm is developed, in which the receive beamforming is updated via the minimum mean square error criterion, while the power allocation and RA rotation are handled by fractional programming and successive convex approximation. Numerical results demonstrate the EE superiority of the proposed RA-NOMA scheme over several benchmarks.

This paper establishes a theoretical framework for the uniform convergence of smoothly activated deep neural network (DNN) estimators. While standard ReLU networks achieve minimax-optimal rates in the $L^2(P)$ norm for various nonparametric regression tasks, we establish a theoretical lower bound demonstrating that least-squares ReLU estimators can suffer from the curse of dimensionality in their uniform convergence behavior. Motivated by the need for reliable uniform guarantees in downstream tasks requiring worst-case reliability, we address this limitation by analyzing smoothly activated DNNs (smooth DNNs), encompassing both feedforward and residual structures. We establish novel pseudo-dimension bounds, non-asymptotic approximation guarantees, and Hölder-norm bounds for the approximators of these models. Leveraging these results, we derive non-asymptotic uniform convergence rates for smooth DNN estimators across multiple statistical contexts, including Huber, least-squares, quantile, and logistic regression. We prove that smooth DNNs can mitigate the {curse of dimensionality} in uniform convergence by adaptively exploiting the low-dimensional hierarchical composition structure of the target function. Supported by both simulation studies and a real-world application, our results position smooth DNNs as a theoretically grounded and practically viable alternative to ReLU networks for statistical learning tasks requiring uniform guarantees.

We solve interface problems on the line between various constant coefficient linear evolution partial differential equations. Our prototypical examples are the heat, linear Schrödinger, Airy, linearized Korteweg de Vries, and biharmonic Schrödinger equations. In each problem, one of the listed equations is posed on one spatial half line, and another on the other half line, with appropriate interface conditions. These problems are solved by means of novel extensions of Fokas's unified transform method and the explicit solution formulae are evaluated using Filon quadrature.

We study a class of nonlinear nonlocal elliptic equations in $\mathbb{R}^N$ involving superpositions of Hartree-type nonlinearities. Motivated by the Schrödinger-Poisson-Slater system, these equations arise as natural generalizations of problems with a single nonlocal interaction term. More precisely, we consider equations driven by a family of Riesz potentials weighted by a positive Borel measure, which gives rise to a superposed nonlocal operator. To treat this problem variationally, we introduce suitable functional settings, namely the superposed Coulomb space and the associated superposed Coulomb-Sobolev space, and study their main properties. Combining variational methods with a recently developed scaling-based critical point theory, we prove existence and multiplicity results for radial solutions. We also investigate a Brezis-Nirenberg-type problem and obtain multiplicity results near eigenvalues of an associated nonlinear eigenvalue problem. Our results extend previous works on single Hartree-type equations and provide a unified framework for treating superpositions of nonlocal interactions of Hartree type.

Despite significant advancements in parameterizations of boundary layer processes, forecasting, and nowcasting low-level clouds using numerical models remain challenging. The purpose of this study is to test a prototype of a high-resolution stochastic-deterministic model designed to simulate the life cycle of fog cover based on the Ising model from statistical mechanics. The case of advection fog around St. John's Airport in Newfoundland (Canada) has been considered. The model demonstrates promising capabilities in forecasting mean fog cover and replicating the horizontal structure observed in satellite imagery, including bands, rolls, and closed or open cells. We evaluate the model's predictive skill by analyzing its effectiveness in reproducing the evolution of fog cover across three representative cases. A contingency table and associated performance metrics are used to assess its accuracy.

We show that two families of operator algebras, the CSL algebras of multiplicity free CSLs and the semicrossed products of commutative C$^*$-algebras, demonstrate a strong form of rigidity with respect to isometric isomorphisms. Specifically, the isomorphism class of any such algebra remains unchanged within its family, even if we allow for isomorphism after tensoring with operator algebras containing the compact operators. For semicrossed products of commutative C$^*$-algebras, the same conclusion holds even when tensoring with operator algebras whose diagonals are irreducibly acting. Collectively, these results imply rigidity with respect to stable isomorphisms: two algebras are isometrically isomorphic if and only if they are stably isomorphic.

We introduce a new topological method to naturally extend a partial function $h \colon X \rightharpoonup [-\infty, \infty]$ on a “generalization” of a metric space $X$ equipped with a dynamical system $f \colon X \rightharpoonup X$, to a function $h_f^{\varepsilon\text{-}\ell^p} \colon X \to [-\infty,\infty]$ with parameters $\varepsilon,p$, which allows us to apply existing topological data analysis techniques to functions defined on the whole space. Moreover, given a function $h$ that evaluates the “quality” of points within $\mathop{\mathrm{dom}}h$, using this extended function, one can construct a sufficient condition for the existence of an optimal $\varepsilon$-perturbation path from any point into $\mathop{\mathrm{dom}}h$ that minimizes the value of $h$ under the condition $X = \mathop{\mathrm{dom}} f \sqcup \mathop{\mathrm{dom}}h = \bigsqcup_{n = 0}^\infty f^{-n}(\mathop{\mathrm{dom}}h)$. In addition, if the domain $X$ is finite, then the function $h_f^{\varepsilon\text{-}\ell^p} \colon X \to [-\infty,\infty]$ can be computed recursively. As an application, for a given partial evaluation function on a space equipped with a dynamical system, one can construct a three-parameter filtration associated with its extension, which naturally identifies minimal paths. This clarifies the relationship among three factors: the evaluation of the cost norm, the strength of control, and the resulting value.

We study the problem of robust repair of a single erasure in Reed–Solomon codes under low communication bandwidth. Focusing on the Guruswami–Wootters trace repair framework, we investigate whether a failed node can be correctly repaired in the presence of erroneous responses from helper nodes. Equivalently, we view the collection of downloaded traces as a code, which we call the repair-trace code. By characterizing the zero coefficients of the associated polynomial in terms of cyclotomic cosets, we derive upper bounds on the dimension $k$ that allow correction of a given number of erroneous traces $e$, as well as lower bounds on the minimum distance as a function of $k$. For the case $q=2$, we exploit explicit formulas for cyclotomic coset representatives to obtain the exact optimal dimension bound for single-error correction. We also propose two efficient robust repair schemes. Our first scheme achieves the error-correction capability guaranteed by the BCH bound. To approach a stronger bound based on character sums, we develop a second scheme that tolerates more errors at the cost of an additional factor $n$ in computational complexity.

Lévy's Dragon Curve is a well-known self-similar fractal, notable for its ability to tile the complex plane. We review a representation of the curve as a set of points given by complex power series satisfying a revolving condition, and study how this representation changes under linear transformations, while preserving its characteristic geometric properties. We introduce a labeled directed graph that encodes these series representations and show that this directed graph remains invariant under linear transformations, with only the labeling subject to change. Furthermore, we provide a geometric characterization of the resulting variations in graph labeling.

Distributional time series arise when each temporal observation is a probability distribution rather than a scalar. We propose Wasserstein exponential smoothing (WES), a one-parameter recursive forecasting method for distributional time series on $\mathbb{R}$. The method adapts the practical logic of classical exponential smoothing to probability distributions by updating forecast distributions along Wasserstein geodesics. This yields a simple filter that can be applied directly to empirical distributions without parametric density modeling. We estimate the smoothing parameter by minimizing an in-sample Wasserstein prediction loss and establish consistency under a distributional local-level data-generating process. In applications to high-frequency equity-index return distributions and household electricity-demand distributions, WES attains the lowest one-step-ahead Wasserstein prediction error among existing distributional autoregressive and regression-based benchmarks for all $20$ series considered, and is retained in the $90\%$ model confidence set in every case.